In the field of signal processing, it is desirable to obtain a 3D wave field mathematical representation of the actual 3D wave field signals as such a representation enables an accurate analysis of the 3D wave field. One such mathematical representation is the 3D wave field spherical harmonic decomposition.
3D wave field signals in a spherical coordinate system (r, θ, φ) can be mathematically represented by equation 1 as an infinite sum of spherical harmonics:
                              P          ⁡                      (                          r              ,              θ              ,              ϕ              ,              k                        )                          =                              ∑                          n              =              0                        ∞                    ⁢                                          ⁢                                    ∑                              m                =                                  -                  n                                            n                        ⁢                                                  ⁢                                          C                nm                            ⁢                                                j                  n                                ⁡                                  (                  kr                  )                                            ⁢                                                𝒫                                      n                    ⁢                                                                m                                                                                            ⁡                                  (                                      cos                    ⁢                                                                                  ⁢                    θ                                    )                                            ⁢                                                E                  m                                ⁡                                  (                  ϕ                  )                                                                                        (                  eq          .                                          ⁢          1                )            where Cnm is the coefficient, jN(kr) is the spherical Bessel function,Ynm=Pn|m|(cos θ)Em(φ) is a representation of the spherical harmonics,
                                          𝒫                          n              ⁢                                              m                                                              ⁡                      (                          cos              ⁢                                                          ⁢              θ                        )                          =                                                                              (                                                            2                      ⁢                                                                                          ⁢                      n                                        +                    1                                    )                                                  4                  ⁢                                                                          ⁢                  π                                            ⁢                                                                    (                                          n                      -                      m                                        )                                    !                                                                      (                                          n                      +                      m                                        )                                    !                                                              ⁢                                    P                              n                ⁢                                                    m                                                                        ⁡                          (                              cos                ⁢                                                                  ⁢                θ                            )                                                          (                  eq          .                                          ⁢          2                )            is the normalized Associated Legendre function, and Em(φ)=(1/√{square root over (2π)})ejmφ is the normalized exponential function. The normalized exponential function represents spherical waves in the φ direction, while the normalized Associated Legendre function represents spherical waves in the θ direction.
The spherical harmonics are orthonormal, therefore satisfying:∫S2Yn′m′(Ω)Y*nm(Ω)d(Ω)=τn-n′τm-m′  (eq. 3)where Ynm=n|m|(cos θ)Em(φ) is a representation of the spherical harmonics.
FIG. 10 shows a plot of the spherical harmonics of order 0 to 3, which shows that the odd-modes 1010 of the spherical harmonics are zero at
      θ    =          π      2        ,and that the even-modes 1020 of the spherical harmonics are non-zero at
  θ  =            π      2        .  FIG. 10 also shows the spherical coordinate system corresponding to the spherical harmonics. The even-modes 1020 are only partially marked in FIG. 10 to avoid cluttering the figure. According to the spherical harmonics, only even-modes 1020 are observable on the x-y plane (i.e.,
  θ  =      π    2  plane). That is, odd modes 1010 are undetectable on the x-y plane. Therefore, sensors need to be placed at different vertical altitudes to acquire the 3D wave field signals in order to fully produce the mathematical representation of the 3D wave field spherical harmonics decomposition.
One type of array configuration fulfilling the above requirement is the spherical array. The geometry of the spherical array coincides with the spherical harmonics, which makes the 3D wave field signals acquired by the spherical array suitable for generating 3D wave field spherical harmonics decomposition. There are two models of the spherical array configuration: the open sphere model (where the sensors are placed in open space) and the rigid sphere model (where the sensors are placed on the surface of a rigid sphere).
A problem however exists with the spherical array in that the array could be ill-conditioned numerically, due to nulls in the spherical Bessel Functions. This problem results in the acquired 3D wave signals being highly sensitive to the diameter of the spherical array. In addition, placement of sensors on the spherical array follows a strict rule of orthogonality of the spherical harmonics, which limits the flexibility of the array configuration (especially in terms of sensor quantity). Further, the spherical shape of the array poses difficulties in regard to implementation as well as being impractical.
Another limitation of the spherical array is the narrow frequency band, due to the nature of the spherical Bessel function. The spherical array is therefore unable to process 3D wave field signals for certain orders of the spherical harmonics. Design of the spherical array must be performed carefully so that the active spherical Bessel functions are non-zero at the spherical array's radius, for the target frequency band.
Thus, a need exists to provide more practical array configurations.